The probabilistic theory of the structure invariants ϕ_h + ϕ_k + ϕ_l + ϕ_m in P

Probability distributions associated with the third, fourth, fifth and sixth neighborhoods of the structure invariant φ_lmφ_h + φ_k + φ_l + φ_m are derived. These distributions yield estimates for φ_lm in terms of magnitudes |E| alone which are particularly good in the favorable case that the standard deviations of the distributions are small. The estimates for the φ_lm lead in turn to unique values for the individual phases φ even for very complex structures. Conditional distributions of φ_lm given the values of one or more structure invariants φ_pq, φ_rs, φ_tu and appropriate sets of magnitudes |E| are also described.